Wavelet MRE: Imaging Propagating Broadband Acoustic Waves with Wavelet-Based Motion-Encoding Gradients

Abstract

Purpose

To demonstrate a novel MR elastography (MRE) technique, termed here wavelet MRE. With this technique broadband motion sensitivity is achievable. Moreover, the true tissue displacement can be reconstructed with a simple inverse transform.

Methods

A wavelet MRE sequence was developed with MEGs based on Haar wavelets. From the phase images displacement was estimated using an inverse transform. Simulations were performed using a frequency sweep and a transient as ground truth motions. A PVC phantom was scanned using wavelet MRE and standard MRE with both transient (one and 10 cycles of 90-Hz motion) and steady-state dual-frequency motion (30Hz and 60Hz) for comparison. The technique was tested in a human brain and motion trajectories were estimated for each voxel.

Results

In simulation, the displacement information estimated from wavelet MRE closely matched the true motion. In the phantom test, the MRE phase data generated from the displacement information derived from wavelet MRE agreed well with standard MRE data. Testing of wavelet MRE to assess transient motion waveforms in the brain was successful and the tissue motion observed was consistent with a previous study.

Conclusion

The uniform and broadband frequency response of wavelet MRE makes it a promising method for imaging transient, multifrequency motion or motion with unknown frequency content. One potential application is measuring the response of brain tissue undergoing low-amplitude, transient vibrations as a model for the study of traumatic brain injury.

Introduction

Standard magnetic resonance elastography (MRE) techniques acquire images of propagating shear waves of a prescribed frequency in steady state (1,2). Several acquisitions are performed using a motion-encoding gradient (MEG) in the shape of one or more cycles of a trapezoidal or sinusoidal waveform, and at multiple, equally spaced phases. Images are acquired representing different phases of the propagating wavefield. As a noninvasive tool for measuring tissue mechanical properties, MRE has an expanding role in clinical applications, from assessing liver fibrosis (3–5) to applications in heart (6,7), brain (8–16), and other organs (1,17–24).

The mechanical properties of biological tissue can be frequency dependent, and there is currently an increasing interest in evaluating such frequency dependency with multifrequency MRE (25–32). In most of the previous studies, MRE images were acquired at each frequency monochromatically. Some MRE techniques have been developed to acquire multifrequency information simultaneously (33–36). These techniques usually make use of the fact that one MEG shape can encode more than one motion frequency (37). However, this approach requires careful planning in terms of MEG shape and frequency as well as the arrangement of the phase offsets. Additionally, motion sensitivity is different for each frequency component, which makes retrieving the ‘true’ waveform a complicated process (37).

Transient waveform MRE is another example of broadband motion detection with increased interest recently (38–43). One application of transient waveform MRE is the study of the brain’s response in the context of traumatic brain injury (TBI). Currently transient brain motion has been studied either using MR tagging techniques (44–46), or estimated using standard MRE with steady-state harmonic motion (47,48), which is quite different from transient motion. Compared to harmonic vibrations, transient motion is a more realistic way of studying how the brain responds to impulsive forces as it more closely resembles the pattern of a real impact. Also, compared to tagging, MRE is better at detecting microscopic (μm) motion and higher motion frequencies, which could provide useful complementary information to MR tagging results (38). However, using standard MRE acquisition techniques to image the propagation of a transient motion impulse is a more challenging problem than multifrequency steady-state MRE, because transient motion has a continuous frequency spectrum of unknown range. Given that there are certain frequencies at which the sinusoidal/trapezoidal MEG has zero sensitivity, it is not always feasible to detect all desired frequency components with reasonable sensitivity. Standard MRE also requires the design of proper deconvolution techniques for transient applications (34).

In this work, we developed a more efficient approach for broadband motion imaging, which we call wavelet MRE. Instead of using a fixed MEG for all phase offsets, wavelet MRE uses multiple scales of wavelet basis functions as MEG shapes for different phase offsets. The advantages of doing so include: 1) multiple scales of wavelet basis functions can easily cover a wide range of frequencies with nearly uniform sensitivity, allowing accurate detection of broadband motion; 2) Wavelet basis functions are orthogonal to each other, which enables more efficient motion encoding and will facilitate future scan time optimization; and 3) The underlying true displacement can be calculated from the MRE phase information with a simple inverse wavelet transform, so deconvolution is no longer necessary. We used Haar wavelets for the MEG profile in this initial work, as Haar is the simplest wavelet system, and its wavelet function is very close to the bipolar MEG used in many MRE applications. Simulation and phantom tests are performed using wavelet MRE and standard MRE for comparison. Finally, this technique is applied in a volunteer study to explore the feasibility of measuring in vivo brain displacements resulting from a low-amplitude transient impact.

Theory

Haar Transform

The Haar system was first proposed in 1909 as an example of an orthonormal system of functions defined on the interval of [0,1] that can be used to approximate any arbitrary function (49). The scaling function, mother wavelet function and the wavelet function in general are described in Equation 1 and shown in Figure 1a.

Figure 1.

(a) Haar wavelet functions; (b) A function (blue line) on [0,1] and 3 scales of Haar wavelets (brown line for ψ(t), green for ψ_1,1(t) and ψ_1,2(t), and grey for ψ_2,1(t), ψ_2,2(t), ψ_2,3(t) and ψ_2,4(t); (c) Inverse Haar transform of the function in (b): brown line from ψ(t) only, grey line from three scale, red line from 10 scales.

$$\begin{gathered} \mathit{\text{Scaling function}}: \phi \left(t\right)=\{\begin{array}{cc}1& 0\le t<1\\ 0& \mathit{\text{otherwise}}\end{array} \\ \mathit{\text{mother wavelet function}}: \psi \left(t\right)=\{\begin{array}{cc}1& 0\le t<1/2\\ -1& 1/2\le t<1\\ 0& \mathit{\text{otherwise}}\end{array} \\ \mathit{\text{wavelet function}}: {\psi }_{j,k}\left(t\right)=\{\begin{array}{cc}{\sqrt{2}}^j& \frac{k-1}{2^j}\le t<\frac{k-1/2}{2^j}\\ -{\sqrt{2}}^j& \frac{k-1/2}{2^j}\le t<\frac{k}{2^j}\\ 0& \mathit{\text{otherwise}}\end{array} \end{gathered}$$ [1]

In these equations, j is the number of scales chosen for analysis and k are the translations at that scale. The Haar series of a given function f(t) on [0,1] is given by Equation 2.

$$\begin{gathered} a_0={\int }_0^{1}f\left(t\right)\phi \left(t\right)dt \\ {a}_1={\int }_0^{1}f\left(t\right)\psi \left(t\right)dt \\ {a}_{j,k}={\int }_0^{1}f\left(t\right){\psi }_{j,k}\left(t\right)dt \end{gathered}$$ [2]

With a₀, a₁ and a_j,k, an inverse Haar transform can be implemented as:

$$f_n\left(t\right)=a_0\phi \left(t\right)+a_1\psi \left(t\right)+\sum _{\begin{array}{c}2\le j\le n\\ 1\le k<2^j\end{array}}{a}_{j,k}{\psi }_{j,k}\left(t\right)$$ [3]

An example of an inverse Haar transform is shown in Figure 1c with n=1, 3 and 10 using the wavelet functions shown in Figure 1b. The scaling function and a₀ were not included so the DC component was not added into the result.

MEG Design

In MRE, the accumulated phase induced by motion in the motion-encoding direction is given by Equation 4:

$$\theta \left(t\right)={\int }_0^t\overrightarrow{G}\left(\tau \right)\cdot \overrightarrow{r}\left(\tau \right)d\tau$$ [4]

where $\overrightarrow{G}\left(t\right)$ is the MEG and $\overrightarrow{r}\left(t\right)$ is the displacement. In standard 3D MRE, $\overrightarrow{G}\left(t\right)=G\left(t\right)\overrightarrow{i}$ for x direction, $G\left(t\right)\overrightarrow{j}$ for y direction, and $G\left(t\right)\overrightarrow{k}$ for z direction. the same G(t) is used for all three directions and for measuring multiple phase offsets or time points of the wavefield using different trigger delays or time shifts between the motion and the MEG. Comparing equations [2] and [4], it is obvious that if, instead of one G(t), a group of $\overrightarrow{G}\left(j,k,t\right)$ were designed in the shape of the Haar wavelet functions, the accumulated phase would be proportional to the Haar transform of r(t) in that direction. Therefore, the actual motion r(t) in each direction can then be estimated with the inverse Haar transform of the MRE phase information.

Normally, one would look at doing a single wavelet decomposition of an entire motion time course using the theory presented above. However, if one were measuring a 200-ms waveform, for example, then one would need 200-ms-long wavelet MEGs, which would make the TE impractically long. So, for implementation into an MRE pulse sequence, the displacement r(t) is separated into multiple sampling windows. The motion that is sampled in the sampling window can be adjusted by starting the motion earlier and earlier before the MEGs (like how conventional MRE samples different phase offsets or time points of a wavefield). In each window, MEGs in the shape of Haar basis functions are used as in Figure 1b. The scaling function encodes the DC component of the function. However, in MRE, the scaling function cannot be used directly as a MEG. Therefore, instead of detecting the DC component of one sampling window, a special MEG was designed to cover two consecutive sampling windows to detect the difference in DC values between the two windows (Figure 2).

Figure 2.

Timeline of the wavelet MRE sequence with eleven offsets in three consecutive sampling windows: black lines = scaling function; red = mother wavelet and green = subsequent wavelets.

For example, if we divide the total time duration of the motion into m windows, each with a length of T; then for window i (0<i<m-1), the MEGs G(j,k,t) can be designed as shown in Equation 5:

$$\begin{gathered} \mathit{\text{Scaling function when}} i>0:G_{i,0}\left(t\right)=G_{\mathit{\text{max}}}\left(\phi \left(\frac{\left(t-iT\right)}{T}+1\right)- \phi \left(\frac{t-iT}{T}\right)\right)=\{\begin{array}{cc}{G}_{\mathit{\text{max}}} \left(i-1\right)T\le t<iT& \left(\mathit{\text{to connect to the previous window}}\right)\\ -G_{\mathit{\text{max}}} iT\le t<\left(i+1\right)T& \left(\mathit{\text{the current window}}\right)\\ 0& \mathit{\text{otherwise}}\end{array} \\ \mathit{\text{mother wavelet function}}: G_i\left(0,0,t\right)=G_{max} \psi \left(\frac{t-iT}{T}\right)=\{\begin{array}{cc}{G}_{\mathit{\text{max}}}& iT\le t<\left(i+\frac{1}{2}\right)T\\ -G_{\mathit{\text{max}}}& \left(i+\frac{1}{2}\right)T\le t<\left(i+1\right)T\\ 0& \mathit{\text{otherwise}}\end{array} \\ \mathit{\text{wavelet function}}: G_i\left(j,k,t\right)=G_{\mathit{\text{max}}} {\psi }_{j,k}\left(\frac{t-iT}{T}\right)=\{\begin{array}{cc}{\sqrt{2}}^j{G}_{\mathit{\text{max}}}& \left(i+\frac{k-1}{2^j}\right)T\le t<\left(i+\frac{k-\frac{1}{2}}{2^j}\right)T\\ -{\sqrt{2}}^j{G}_{\mathit{\text{max}}}& \left(i+\frac{k-\frac{1}{2}}{2^j}\right)T\le t<\left(i+\frac{k}{2^j}\right)T\\ 0& \mathit{\text{otherwise}}\end{array} \end{gathered}$$ [5]

The accumulated phase in one motion-sensitization direction is described in Equation 6:

$$\begin{gathered} {\theta }_{i,0}=\gamma {\int }_{\left(i-1\right)T}^{\left(i+1\right)T}r\left(t\right)G_{i,0}\left(t\right)dt=\gamma G_{\mathit{\text{max}}}{\int }_{\left(i-1\right)T}^{\left(i+1\right)T}r\left(t\right)\left(\phi \left(\frac{t-iT}{T}+1\right)- \phi \left(\frac{t-iT}{T}\right)\right)dt={\theta }_{i-1,0}-\gamma T{G}_{\mathit{\text{max}}}{a}_{i,0} (\mathit{\text{when}} i>0) \\ {\theta }_i\left(0,0\right)=\gamma {\int }_{iT}^{\left(i+1\right)T}r\left(t\right)G_i\left(0,0,t\right)dt=\gamma G_{\mathit{\text{max}}}{\int }_{iT}^{\left(i+1\right)T}r\left(t\right)\psi \left(\frac{t-iT}{T}\right)dt=\gamma T{G}_{\mathit{\text{max}}}{a}_{i,0,0} \\ {\theta }_i\left(j,k\right)=\gamma {\int }_{iT}^{\left(i+1\right)T}r\left(t\right)G_i\left(j,k,t\right)dt=\gamma G_{\mathit{\text{max}}}{\int }_{iT}^{\left(i+1\right)T}r\left(t\right){\psi }_{j,k}\left(\frac{t-iT}{T}\right)dt=\gamma T{G}_{\mathit{\text{max}}}{a}_{i,j,k} \end{gathered}$$ [6]

With the phase of the images acquired using n scales of Haar wavelets, the displacement in the ith window can finally be reconstructed as described in Equation 7.

$$r_{i,n}\left(t\right)=\frac{1}{\gamma T{G}_{max}}\left(\left({\theta }_{i-1,0}-{\theta }_{i,0}\right)\phi \left(\frac{t-iT}{T}\right)+{\theta }_i\left(0,0\right)\psi \left(\frac{t-iT}{T}\right)+\sum _{\begin{array}{c}2\le j\le n\\ 1\le k<2^j\end{array}}{\theta }_i\left(j,k\right){\psi }_{j,k}\left(\frac{t-iT}{T}\right)\right)$$ [7]

Methods

Simulation

A simulation was performed to verify the theory described above using MATLAB (MathWorks, Natick, Massachusetts, USA). Two ‘true’ motion curves were used with a time step of 0.01 ms and a duration of 245.75ms: a frequency sweep and a transient. The frequency sweep had a frequency range of 20Hz - 80Hz. The transient motion was calculated from the detected acceleration measured in a phantom, using an accelerator (Model #3145–002, ICSensors Inc., Milpitas, CA) on top of a PVC phantom with one cycle of 90Hz motion applied with a MRE driver. This 245.75ms interval was divided into 6 sampling windows for reconstructing the original motion and for estimating the frequency response of the system. Both idealized rectangular Haar wavelet MEGs and more realistic trapezoidal approximations (slope length = 0.5ms) were used in the simulations. Due to the temporal resolution (0.01 ms) in this simulation, the wavelet transform can be performed with up to 12 scales. Due to the limit of the slope length, only 4 trapezoidal wavelet scales (and therefore also only 4 rectangular wavelet scales) were used for reconstructing the real motion from the simulated MRE phase information. For comparison, a simulation of standard MRE was also performed with a standard trapezoidal, 60-Hz, bipolar MEG (duration 16.67 ms), and a trigger delay of 4 ms.

In addition, with both rectangular and trapezoidal MEGs the frequency response of the system was estimated by taking the Fourier transform of the impulse response.

Pulse Sequence

A wavelet MRE sequence was developed based on a spin-echo EPI MRE sequence with 3D motion encoding. Figure 2 shows the timeline of one motion encoding direction with three sampling windows and two scales of wavelet. An external trigger was sent from the scanner to the active driver prior to the RF of each TR by a certain trigger delay. Different motion encoding gradients were implemented for each phase offset. The first group is for the scaling function, in which bipolar MEGs were used with length equal to twice the window length and each phase offset had an additional trigger delay of one window length, as shown in phase offsets 1–2 (the blue TRs) in Figure 2. The second group is for the mother wavelet, with bipolar MEG length equal to the window length and with an additional trigger delay of the window length (phase offsets 3–5, the red TRs in Figure 2). The third group is for the next scale of wavelet, with bipolar MEG length equal to half of the window length and with a trigger delay equal to this MEG length (phase offsets 6–11, the green TRs in Figure 2). This pattern can continue for further scales of wavelets as needed (not shown): each additional scale would have an MEG length half that of the previous group and phase offsets with a trigger delay equal to that MEG length. All these phase offsets are implemented in three motion encoding directions for 3D applications.

Phantom Test with Transient Motion

Data were acquired on a 3T clinical scanner (MAGNETOM Skyra, Siemens Healthcare, Erlangen, Germany) with a PVC gel phantom. Three scales of Haar wavelets plus a scaling function were used for the MEGs in wavelet MRE. For the scaling function, a 36-ms, 10-mT/m, bipolar MEG detected the difference in the DC value between consecutive 18-ms windows and resulted in a TE of 85ms. MEGs with widths of 18ms, 9ms and 4.5ms were used for the 3 wavelet scales with MEG amplitudes of 10, 14.1 and 20mT/m, respectively. Ten sampling windows were used, and a total of 79 phase offsets were acquired including 9 for scaling function, 10 for G(0,0,t), 20 for G(1,1,t) and G(1,2,t), and 40 for G(2,1,t) to G(2,4,t). The total sampled time duration was 180 ms. For comparison, standard MRE images were also acquired using a 50-Hz MEG and 60 offsets, with TE = 80ms. Common sequence setting was: one slice, TR = 2000ms, FOV = 240×240mm², slice thickness = 3mm and MEG ramp time = 0.47ms. Phase differences were obtained for both techniques.

Two types of transient motion waveforms were generated using a pneumatic driver system (Resoundant Inc., Rochester MN, USA) for this study: (1) one cycle of a 90-Hz sinusoidal wave, and (2) ten cycles of 90-Hz sinusoidal waves. Vibration was generated before each TR as shown in Figure 2 and propagated to the phantom in about 30ms.

To compare the wavelet MRE to standard MRE, the calculated displacement was interpolated to a sampling rate of 100 samples/ms and cross-correlated with a 50-Hz MEG (as was used in the standard MRE) to produce simulated phase information θ_n(t):

$${\theta }_n\left(t\right)={\int }_0^t\overrightarrow{G}\left(\tau \right)\cdot \overrightarrow{r_n}\left(\tau \right)d\tau$$ [8]

The simulated phase profile was then compared with the phase data acquired with standard MRE (also interpolated).

Phantom Test with Steady-State Motion at Multiple Frequencies

To demonstrate how wavelet MRE can also recover the wave information for steady-state MRE studies, a motion waveform was generated that was a combination of continuous 30-Hz and 60-Hz sinusoidal waves with equal amplitude. For wavelet MRE, a total 36 ms of motion was separated into 2 windows, with two wavelet scales in addition to the scaling function so in total 7 phase offsets were acquired. The MEG amplitude was 5mT/m and 7mT/m, and the TR/TE = 3200ms/47ms. With standard MRE, 8 phase offsets over 33.3ms were acquired. MEG frequency = 120Hz, MEG amplitude = 40mT/m, and TR/TE = 3200ms/35ms. The two frequency components were separated using a Fourier Transform after the images were acquired. The ratio between the 30-Hz and 60-Hz motion amplitudes was calculated.

Volunteer Study

To demonstrate how wavelet MRE can recover the motion information for in vivo MRE studies, a healthy volunteer was scanned on a 3T clinical scanner (MAGNETOM Prisma, Siemens Healthcare, Erlangen, Germany). This study was approved by the Institutional Review Board and informed consent was obtained. One cycle of a 90-Hz sinusoidal motion was generated by a pneumatic driver system (Resoundant Inc., Rochester MN, USA) at the onset of each TR. A passive pillow-like driver was placed under the subject’s head. Ten consecutive sampling windows of 10ms were sampled for a total motion sampling time duration of 100ms. Two wavelet scales were used in addition to the scaling function which resulted in 39 phase offsets in total. The gradient amplitudes were 40, 40, and 56mT/m, respectively. The TR/TE was 1000/55ms. One sagittal slice was acquired with a slice thickness of 3mm and in-plane resolution of 3×3mm² and FOV = 240×240mm². The total scan time was about 6 minutes. Displacement was estimated using the inverse Haar transform of the phase images. 3D displacement trajectories were plotted for each voxel. The displacement vectors in the brain were fitted to a rigid body model to estimate the rigid-body translation and rotation (50). The 2D strain tensor was calculated to estimate the brain deformation. The 3D motion of each voxel was also projected onto the magnitude images for animated visualization of brain motion during the transient impact.

Results

Simulation

Figure 3a shows the frequency sweep (used as ‘true motion’), and the detected phase obtained by the standard 60-Hz MEG. The detected phase (blue line) amplitude was low when the motion was around 20Hz, increased when the frequency increased and reached the maximum at around 60Hz; and then decreased again and became unstable when the motion frequency reached 80Hz. These results indicates that the motion sensitivity of the standard 60-Hz MEG is lower at the ends of the frequency range and higher at the frequencies around 60 Hz. In addition, a phase delay occurred at higher and lower frequencies. Using wavelet MRE (Figure 3b), on the other hand, with 4 scales of the rectangular Haar wavelets, the reconstructed results (cyan line) followed the true motion with no phase delay. The motion amplitude was slightly lower at low frequency and decreased a little more when the frequency increased. With the trapezoidal Haar wavelet (green line), the detected motion amplitude further decreased, especially at higher frequencies, but the waveform was still well preserved with no phase delay (Figure 3b). Figure 3c and d shows the simulation results using the measured transient motion curve as ‘true motion’. In Figure 3c, with a standard 60Hz MEG, the detected phase curve shows an amplitude variation and phase delay compared with the ‘true motion’. In Figure 3d the reconstructed displacement from 4 scales of Haar wavelet follows the ‘true motion’ very closely from the beginning to the end. The amplitude of the displacement from trapezoidal MEG is slightly lower than the ‘true motion’ amplitude at the onset of the motion, but after two peaks almost overlaps with the ‘true motion’ and the curve from rectangular Haar wavelets.

Figure 3.

Simulation with a predefined frequency-sweep motion (a,b) or transient motion (c,d) (red line) as ground truth. (a,c) The phase accrual (blue line) from standard MRE with a 60-Hz MEG, and (b,d) the reconstructed displacement from wavelet MRE and the inverse Haar transform using 4 levels of MEGs in the shape of ideal Haar wavelets (blue line) and trapezoidal wavelets (green line).

Figure 4 shows the frequency response of the wavelet systems. With the ideal rectangular Haar system, the transform/inverse-transform pair behaves like an all-pass filter with all 12 levels of Haar wavelets; while with 4 scales (from a width of 40.96ms down to 5.12ms) it becomes a low-pass filter with gradually attenuating amplitude response with the increase of frequency (Figure 4b). Figure 4c shows that the cutoff frequency (3dB attenuation) is at 146Hz. With the trapezoidal approximation of the Haar wavelets with 4 scales, the frequency response still acts as a low-pass filter, but with a slightly narrower passband than the ideal rectangular case (Figure 4d,e), with a cutoff frequency (3dB attenuation) at 139Hz. The 12-level trapezoidal wavelet system also acts as a low-pass filter with a cutoff at about the same frequency as with 4 scales.

Figure 4.

Frequency response of the Haar system. (a) Haar wavelet functions. Frequency response of the ideal Haar system (b, c) and trapezoidal Haar system (d, e) from scaling function only (blue line), one scale of wavelets (red line), 4 scales (green line), and 12 scales (black line). (b, d) show plots with linear axis, (c, e) show magnitude in dB and frequency with logarithm axis.

Phantom Test with Transient Motion

The phase-difference images acquired with each MEG scale (Figure 5a) formed wave images highlighting different wavelengths (Figure 5b). Figure 5b shows the 2^nd offset with 36ms MEG, the 3^rd offset with 18ms MEG, the 5^th offset with 9ms MEG and the10^th offset with 4.5ms MEG. Figure 5c showed the reconstructed motion using these images, corresponding to the motion at ~42ms after the onset of motion. The displacement maps were reconstructed by performing an inverse Haar transform of the phase-difference data (Figure 5c). The estimated displacement at one voxel (yellow dot in Figure 6a) induced by a single 90-Hz wave cycle appeared to oscillate more slowly than 90 Hz (Figure 6b), but with ten 90-Hz wave cycles, the transient motion ultimately reached a steady 90 Hz vibration (Figure 6c). Figure 7 shows that the phase differences reproduced from wavelet MRE agree very well with the acquired phase differences from standard MRE, especially when the phase difference peak was above ~0.4 rad. The phase difference reproduced from wavelet MRE measured displacement had a slightly higher amplitude in all three directions for both 1 and 10 cycles of 90Hz vibration, indicating a systematic bias in one or both MRE techniques. From a voxel-by-voxel linear regression between these two phase differences, correlation coefficient (r) maps were obtained for all three motion directions and all voxels with a phase difference peak above 0.4 rad (Figure 8). The correlation coefficient in the y direction was above 0.95 everywhere in the phantom; while the correlation coefficients in the x and z directions (with much lower motion amplitude) were between 0.7 to 1 for most of the voxels in the phantom.

Figure 5.

(a) MEGs used in the acquisition including three wavelet scales and the scaling function. (b) The phase-difference images acquired with these 3 scales and the scaling function. (c) The displacement reconstructed using the inverse Haar transform (at around 42ms after the trigger).

Figure 6.

Displacement curves estimated at location of the yellow dot in (a) induced by (b) one cycle of 90Hz and (c) 10 cycles of 90 Hz.

Figure 7.

Phase difference reproduced from the reconstructed displacement (black lines) vs. the phase difference acquired using standard MRE (red lines) in the x (a, b), y (c, d) and z (e, f) direction for motion induced by one cycle of 90-Hz motion (a, c, e) and 10 cycles of 90-Hz motion (b, d, f).

Figure 8.

Correlation coefficient (r) maps between the phase reproduced with wavelet MRE measured displacement and the phase acquired using standard MRE, for all voxels with a phase difference peak over 0.4 rad. (a) and (b) show the maps in x direction with the motion of (a) 1 cycle of 90Hz, (b) 10 cycles of 90Hz; (c) and (d): Maps in y direction with motion of (c) 1 cycle of 90Hz, (d) 10 cycles of 90Hz; (e) and (f): Maps in z direction with motion of (e) 1 cycle of 90Hz and (f) 10 cycles of 90Hz.

Phantom Test with Steady-State Motion at Multiple Frequencies

Both wavelet and standard MRE techniques detected both frequency components of the composite 30-Hz and 60-Hz motion (Figure 9). The displacement map reconstructed from wavelet MRE shows overlapping long and short wavelength patterns (Figure 9a). Figure 9b shows the ratio map of amplitude of 60-Hz wave component over that of the 30-Hz wave component (motion direction shown in the white arrow). The average of this ratio is 0.44. Figure 9e shows the phase difference map from standard MRE. Compared with Figure 9a, 9b shows a more prominent short-wave pattern. This is because the standard MRE is more sensitive to 60-Hz motion due to the frequency response of the selected MEG, with a sensitivity ratio between 30-Hz and 60Hz of 1:3.27. The detected signal amplitude ratio of 60-Hz vs. 30-Hz motion in the phantom was 1.38 with standard MRE (Figure 9f). After the standard MRE results were adjusted by the sensitivity for each frequency component, the detected motion amplitude ratio was 0.42 from standard MRE. Therefore, the composite wave imaging results from the two techniques were consistent. The stiffness calculated from the 30-Hz wave images was 2.4±0.3kPa with wavelet MRE and 2.3±0.2kPa with standard MRE (Figure 9c and 9g). The stiffness calculated from the 60-Hz motion was 2.7±0.3kPa with wavelet MRE and 2.4±0.2kPa with standard MRE (Figure 9d and 9h).

Figure 9.

(a) Displacement reconstructed from the wavelet MRE with 30-Hz and 60-Hz composite steady-state motion. (b) Wave amplitude ratio (60-Hz/30-Hz motion) from (a). (c) Stiffness from (a) using the 30-Hz motion. (d) Stiffness from (a) using 60-Hz motion. (e) Phase-difference images from standard MRE. (f) Wave amplitude ratio (60-Hz/30-Hz motion) from (d). (g) Stiffness from (d) using the 30-Hz motion. (h) Stiffness from (d) using 60-Hz motion.

Volunteer Study

Figure 10a shows the magnitude image of the acquired MRE data. Figure 10b shows the motion trajectory of four representative voxels in the brain. The trajectories closely resemble the ‘figure-8‘ trajectory reported in previous studies (51). The analysis of the decomposed translational and rotational motion (Figure 10d,e) indicates that the rigid body response started at a higher frequency and amplitude, then both the amplitude and frequency decreased during the 100-ms time window. The peak frequency during the whole 100-ms window was approximately 30Hz. Translational motion reached its peak at 35.9 ms with an amplitude of ~29μm (Figure 10d), while the rotational motion peaked later at 38.5 ms with an amplitude of ~11×10⁻⁵ degree (Figure 10e). The strain maps across the brain showed a peak at 46.2 ms, which notably lagged the peaks in rigid-body motion (Figure 10c). All the strain maps over the entire time window are available in Figure S1.

Figure 10.

(a) Magnitude image of the volunteer data; (b) 3D motion trajectories of four voxel reconstructed from wavelet MRE; (c) strain maps at peak; (d) translational displacement of the whole brain; and (e) rotational motion of the whole brain.

As further demonstrated in supporting Video S1 (with the motion amplified 500 times), during the impact induced by the brain MRE driver, the initial bulk movement of the brain was toward the contrecoup location (i.e., front) with subsequent movement toward the coup location (i.e., back), which is consistent with the classic coup and contrecoup injury pattern. Of note, when the brain first encountered the inner skull, shear motion was generated in all three directions. The brain then quickly moved back, with the shear wave starting to propagate through the whole brain. Video S2 shows the shear motion in the brain after the bulk motion was removed.

Discussion and Conclusions

A wavelet MRE technique was developed to characterize broadband tissue motion using MEGs in the shape of wavelets with multiple scales. The sampling time window and the number of scales can be optimized for the specific motion of interest. We used transient motion and dual-frequency motion as examples, each tested with the sampling window and number of scales appropriate for each application. We expect that such a motion encoding gradient design strategy can be used with other broadband motion as well.

The simulation results confirm the theoretical prediction that this technique can detect broadband motion with a more uniform frequency response than standard MRE. The cross-correlation between the displacement from wavelet MRE and the MEG function (i.e., the simulated standard MRE phase signal) matched very well with the actual phase data obtained using standard MRE, indicating high consistency between the two MRE techniques within the bandwidth of a 50Hz standard MEG. Compared to standard MRE, the phase differences acquired from wavelet MRE were slightly higher than those from standard MRE. This may be caused by the difference in MEG shape (bipolar for wavelet MRE and conventional flow-compensated 1–2-1 for standard MRE) and gradient nonlinearity effects between the two methods.

In the steady state multifrequency test with a phantom, while the composite waveform that was input to the acoustic driver system had equal 30-Hz and 60-Hz amplitudes, the amplitude ratio of the 30-Hz and 60-Hz waves in the phantom was expected to be something other than 1 due to differences in the response of the driver system and the phantom to different vibration frequencies. The stiffness measurements using wavelet MRE and standard MRE were very close (51,52). The ratio of two frequency components can be directly measured using wavelet MRE. With standard MRE, however, this ratio needs to be calculated based on the sensitivity of MEG on each frequency component.

In the volunteer test with transient motion, wavelet MRE was able to measure broadband transient motion in the brain in vivo, and the estimated motion trajectories were consistent with results from previous studies (51). The strain maps exhibited partial similarity to those acquired via MR tagging during an occipital impact (53).

As the initial proof-of-concept study for this new technique, there are several limitations. One limitation in the sequence design is the selection of MEG shape. In this study, the Haar wavelet system was selected for the MEG profiles in the MRE sequence because it enables local spectral and temporal information extraction, and its shape can be easily approximated by predefined trapezoidal gradient pulses. Our results showed that the use of the trapezoidal approximation as opposed to ideal Haar wavelets did not strongly affect the motion detected. However, the MEG profiles could potentially be optimized by using other wavelet basis functions which may be smoother (54), easier for the gradient coils to realize, or more efficient for encoding tissue motion.

Another limitation is in the validation test. When comparing wavelet MRE with standard MRE in the phantom test using transient motion, we acquired only one set of standard MRE data with an MEG of 50Hz. While this served well as an initial validation of wavelet MRE, to test its broadband sensitivity, a more thorough evaluation should involve comparing wavelet MRE with several standard MRE scans with various MEG lengths (e.g., 20–200Hz).

Harmonic motion with multiple frequency components is another potential application of this new technique. One of the limitations in this work is that we used a dual-frequency motion for the test, so the frequency range may not be ‘broad’ enough to demonstrate the potential of the technique. Motion with more frequency components may be needed in a future study for a full validation.

The broadband nature of this method makes it potentially helpful in measuring transient motion in which the motion bandwidth is not only very broad but also unknown. One potential application could be measuring transient brain motion in the study of brain trauma. Knowledge of in vivo brain motion under a transient impact would help elucidate the mechanisms by which head trauma causes hemorrhage, cerebral contusions, and diffuse axonal injury. Previously, standard MRE with steady-state motion was used in combination with MR tagging to study the brain motion when an impact happens (44–48). This new technique, which detects low amplitude and relatively high frequency motion, can provide another choice for the estimation of the transient motion.

The next step will be to compare the motion-encoding efficiency of this new technique to standard MRE in multifrequency motion detection and in transient motion detection. With standard MRE, the shape of the motion-encoding gradient, the number of phase offsets, and the deconvolution algorithm can be adjusted. A comparison can then be performed between the two MRE techniques in acquisition time and the reconstructed motion. More studies in transient brain motion detection will be conducted in healthy volunteers to further validate its application in this area. Furthermore, optimizations of wavelet transient MRE inversion algorithms, motion analysis and pulse sequence will be performed with the aim of 3D volume data acquisition and inversion within a clinically feasible time.

In summary, these initial results indicate the potential of wavelet MRE for assessing broadband tissue motion. The reproduced phase data, motion amplitude ratio between difference frequency components and the measured stiffness were all consistent with standard MRE. Wavelet MRE could be a promising technique for detecting broadband/transient tissue motion and may have applications in tissue biomechanical studies related to traumatic brain injury.

Supplementary Material

Fig S1

Figure S1, strain inside brain within the sampling window.

Video S1

Video S1. Head motion under impact (x500).

Video S2

Video S2. Brain shear motion (after rigid body motion removed, x500).